To graph the rational function \( f(x) = \frac{(x+4)(x-2)}{x^2-4} \), we need to analyze and determine several key features of the function. Follow these steps:
### Step 1: Simplify the Equation
First, simplify the given function if possible:
[tex]\[ f(x) = \frac{(x+4)(x-2)}{x^2-4} \][/tex]
Notice that the denominator can be factored:
[tex]\[ x^2 - 4 = (x-2)(x+2) \][/tex]
So, the function becomes:
[tex]\[ f(x) = \frac{(x+4)(x-2)}{(x-2)(x+2)} \][/tex]
For \( x \neq 2 \) and \( x \neq -2 \):
[tex]\[ f(x) = \frac{x+4}{x+2} \][/tex]
### Step 2: Identify Domain Restrictions
The original function has restrictions due to the denominator:
[tex]\[ x^2 - 4 \neq 0 \implies x \neq 2 \text{ and } x \neq -2 \][/tex]
These values create vertical asymptotes. Additionally, factor cancellation makes these places where \( f(x) \) is undefined without corresponding holes in the graph.
### Step 3: Vertical Asymptotes and Holes
Since the \( x-2 \) term cancels in the numerator and denominator, it creates a hole at \( x = 2 \):
- Hole at \( x = 2 \)
Plug \( x = 2 \) into the simplified form:
[tex]\[ f(2) = \frac{2+4}{2+2} = \frac{6}{4} = 1.5 \][/tex]
So, there's a hole at \( (2, 1.5) \).
The vertical asymptote is at \( x = -2 \).
### Step 4: Find the Horizontal Asymptote
Examine the degrees of the polynomials in the numerator and the simplified denominator:
[tex]\[ \frac{x+4}{x+2} \][/tex]
Since both the numerator and denominator are of the same degree (1), the horizontal asymptote can be found by dividing the leading coefficients:
[tex]\[ \text{Horizontal asymptote: } y = \frac{1}{1} = 1 \][/tex]
### Step 5: Determine Intercepts
Y-intercept: Set \( x = 0 \):
[tex]\[ f(0) = \frac{0+4}{0+2} = \frac{4}{2} = 2 \][/tex]
So, the y-intercept is \( (0, 2) \).
X-intercepts: Set \( f(x) = 0 \):
The rational fraction is zero when the numerator is zero, i.e., \( x + 4 = 0 \):
[tex]\[ x = -4 \][/tex]
So, the x-intercept is \( (-4, 0) \).
### Step 6: Plot Points and Sketch the Graph
Now, let's gather this information to plot the graph:
1. Hole: at \( (2, 1.5) \) (draw a small open circle to indicate the hole)
2. Vertical Asymptote: \( x = -2 \) (draw a dashed vertical line)
3. Horizontal Asymptote: \( y = 1 \) (draw a dashed horizontal line)
4. Y-intercept: \( (0, 2) \)
5. X-intercept: \( (-4, 0) \)
Sketch the graph with these features in mind, approaching the asymptotes but never crossing them, and noting the hole at \( x = 2 \).
By plotting these points and asymptotes and drawing the graph smoothly according to these guidelines, you will have accurately graphed the function [tex]\( f(x) = \frac{(x+4)(x-2)}{x^2-4} \)[/tex].
